An electronic motor drives a machine through a speed$-$reducing gear ratio $9$:$1.$ The motor armature, with its shaft and gear wheel, has a moment of inertia $0\mathrm{.}5kg{m}^2.$ The rotating part of the driven machine has a moment of inertia of $40kg{m}^2.$ The driven machine has a constant resisting torque of $100$ Nm and the efficiency of the reduction gear is $95\%.$

a$.$ What power must the motor develop to drive the machine at a uniform speed of $160re\frac{v}{m}in?$

b$.$ The torque developed on the motor armature, in starting from rest, is $30$ Nm $.$ What time will be required for the speed of the machine to increase from $0$ to $60re\frac{v}{m}in?$

c$.$ If the Gear ratio were altered so as to give the machine the greatest possible angular acceleration in starting from rest, what will then be the gear ratio? The starting torque of the motor is $30$ Nm as before.